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WHAT IS MODERN IN “MODERN MATHEMATICS”?
HOW SHOULD MODERN TEACHING REFLECT THIS?
Imre BOKOR
School of Mathematics, Statistics and Computer Science
University of New England
Armidale, 2351 New South Wales
Australia
e-mail: [email protected]
ABSTRACT
It is commonly held that what distinguishes modern mathematics is the availability of highspeed electronic computers and pocket calculators with graphical capabilities. Consequently,
mathematics is usually taught in schools and undergraduate courses as if Euler and Gauss
were our contemporaries, with electronic gadgets replacing tables, slide-rules and sketching.
We discuss cultural changes in mathematics over the past two hundred years overlooked
by such an approach, specifically the rise of rigour and algebra. These have altered the face
of mathematics, providing a deeper understanding of many important results, by providing a
unified, coherent setting.
At the same time, these developments have increased the power and scope of mathematics, by enabling it to deal with non-quantitative problems, by making many computations
accessible to computers and by making it more applicable to other disciplines.
We use, in particular, the Fundamental Theorem of Calculus as an illustration and offer
a programme for teaching calculus in a manner which accommodates these developments and
eases the student’s path to further studies.
1
Background
The purpose of this paper is to provide a sketch of cultural changes in mathematics over
the past two hundred years, using in particular the Fundamental Theorem of Calculus
as illustration. Brevity dictates that details be missing.
Such discussion as this paper intends to foster is urgently needed, for at the dawn of
the 21st century undergraduate and high school mathematics are generally still taught
as if Gauss and Euler were contemporaries, rather than historical figures whose bicentenaries have been all but forgotten.
Two main strands are discernible in mathematics, often perceived to be in conflict,
namely solving problems and constructing theories. The history of mathematics clearly
shows they are symbiotic. For trying to solve previously intractable problems has frequently led to advances in available theory or the development of new theory. Newton’s
Differential Calculus is a prime example. Equally, purely theoretical advances have
frequently — and unanticipatedly — led to solutions of problems in areas to which no
significant connection had been suspected. The application of the theory of fibre bundles to arbitrage, of cohomology to number theory and of Hopf algebras to theoretical
physics (in the guise of quantum groups) spring immediately to mind. A propos the last
example, in [2], Dieudonn´e was still able to write of the connections with the natural
sciences of category theory (p.246) and homological algebra (p.180) “None at present”.
How dramatically and quickly that changed!
Moreover, we have witnessed the same theory has arise independently and contemporaneously from both practical needs and “purely speculative” considerations. For
example, Russell and Whitehead’s philosophical programme in Principia Mathematica
and Dehn’s interest in knots both led to the word problem in group theory.
However, this symbiosis does not mean that there is no clearly discernible trend in
the history, development and culture of mathematics.
The overriding trend in mathematics over the last two centuries has been increasing
rigour and increasing “algebraisation”: There has been a systematic formalisation and
axiomatisation of calculations and arguments. This is not due to a theological addiction
to formalism, but the inevitable consequence of several developments: Disturbing paradoxes were found to be immanent in mathematics as practised, and several cherished
expectations were dashed by the cold light of irrefutable reason.
The discovery of non-Euclidean geometry by Bolyai, Lobachevski and Riemann,
the proof by Galois of the unsolvability of polynomial equations of degree higher than
four and the discovery of the paradoxes of set theory shook mathematics to its very
foundation, destroying the cherished certitude of established beliefs and confident expectations.
New phenomena have been observed which were unthinkable observed without the
guidance of mathematically formulated theory. The discrepancy between the prediction
from our theoretical model and the actual recorded observations of our planetary system
led to the postulation of previously undiscovered planets, and the subsequent search,
guided by the theoretical predictions, resulted in the discovery which confirmed the
theory. Who would have thought of trying to measure the bending of light rays around
Mercury had Einsteinian theory not predicted it?
Mathematics had provided coherent explanation of events, processes and observations in terms of elegant and insightful theory.
Instead of abandoning mathematics, a rigorous, axiomatic formulation was sought,
devoid of the pitfalls while preserving the powerful theories and theorems. By and
large, the scientific and philosophical community was convinced that the major results
were correct, even though some of the reasoning used to justify them was flawed.
So, after centuries of primary concern with problem-solving, mathematics was forced
to concern itself increasingly with constructing theories, not just in an ad hoc manner
to justify particular techniques to solve particular problems, but in a considerably more
serious and catholic manner.
An enduring legacy of this development is that by becoming more fundamental,
mathematics today is more abstract and rigorous, thereby becoming more applicable
and more broadly applied. Electronics, meteorology, modern “financial management”,
not to speak of computing and computers, quantum theory and relativity theory are
all inconceivable in their current forms without modern mathematics.
Paul Dirac’s observation in 1931 ([3] p. 368) has lost nothing of its aptness.
“The steady progress of physics requires for its theoretical formulation
a mathematics that gets continually more advanced. This is only natural
and to be expected. What however was not expected by the scientific workers
of the last century was the particular form that the line of advancement of
the mathematics would take, namely, it was expected that the mathematics would get more and more complicated, but would rest on a permanent
basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundations
and gets more abstract. Non-euclidean geometry and non-commutative algebra, which were at one time considered to be purely fictions of the minds
and pastimes of logical thinkers, have now been found to be very necessary
for the description of general facts of the physical world. It seems likely that
this process of increasing abstraction will continue in the future and that
advances in physics is to be associated with a continual modification of the
axioms at the base of mathematics rather than with a logical development of
any one mathematical scheme on a fixed foundation.”
2
The Fundamental Theorem of Calculus Revisited
The Fundamental Theorem of Calculus serves well to illustrate Dirac’s point.
If f : [a, b] −→ R is continuous, then the function
Z x
F : [a, b] −→ R, x 7−→
f (t) dt
(1)
a
is continuous on [a, b] and differentiable on ] a, b [ with
F 0 (x) = f (x)
(2)
for a < x < b.
This was originally an astounding theorem, for it demonstrated that two apparently
unrelated problems — finding a function whose derivative is a given function and finding
the average value of a given function — have a common solution. It also provided a
link between the then new differential calculus, and integration, which had been known
since antiquity in the guise of the “method of exhaustion”.
One immediate consequence is the use of results from differential calculus to provide
strategies and techniques for integral calculus, as we discuss below.
One direction in which calculus subsequently developed is multivariate calculus,
from which we quote some results — Stokes’ and Gauss’ theorems — in a convenient
form.
Let F : R3 −→ R3 be differentiable. Let S be a suitable oriented surface in R3 with
boundary ∂S, which we take with the induced orientation. Let V be a suitable oriented
solid region of R3 with boundary ∂V , which we take with the induced orientation. Then
Z
Z
F · dr =
∇ × F · ds
(3)
∂S
S
Z
Z
∇ · Fdv
(4)
F · ds =
∂V
V
where r, s and v denote line, surface and volume elements.
Thinking of F 0 as the gradient of F , and agreeing that integrating a function on a
finite set consists of summing its values on that set, we obtain a reformulation of the
Fundamental Theorem of Calculus.
Let F : R −→ R be differentiable. Let I be an oriented interval whose boundary ∂I
is taken with the induced orientation. Then
Z
Z
F = ∇F · dr.
(5)
∂I
I
Thus, Green’s, Stokes’ and Gauss’ theorems are merely “higher dimensional” —
more “complicated”, as Dirac would say — versions of the Fundamental Theorem of
Calculus. Alternatively we may view the Fundamental Theorem of Calculus as a special
case or application of the others.
Subsequent developments took a different direction, with far more profound consequences. They consisted of the investigation, primarily due to Poincar´e, of the word
“suitable” in the above theorems. Poincar´e studied simplexes and their boundaries. A
k-simplex in Rn is the convex hull of k + 1 points. However the boundary of a simplex
is not a simplex, but rather a chain of simplexes.
Writing σ for a simplex, ∂σ for its boundary, ω for a differential form and dω for
its differential, the above theorems all assume the form
Z
Z
ω =
dω.
(6)
∂σ
σ
[Recall that a differential 0-form is just a smooth real valued function f (x1 , . . . , xn )
and its differential is the 1-form
X ∂f
df :=
dxj
(7)
∂x
j
j
If ω = f dxj1 . . . dxjk is a differential k-form, then its differential is the (k + 1)-form
dω :=
X ∂f
dxi dxj1 . . . dxjk
∂x
i
i
(8)
with the convention that dxdx = 0.]
The work initiated by Poincar´e involved the notions of homology, homotopy and
Betti numbers to describe regions of Rn and he introduced a group, the fundamental
group, to characterise when the integrals of a function along two paths between the
same two points necessarily coincide.
As a result of the insights and influence of Emmy Noether, these notions were
recognised to be best formulated in terms of homology, cohomology and homotopy
groups.
Cartan was able to provide a sound algebraic foundation for what physicists and
engineers had been practising with differential forms, often to the consternation of
mathematicians.
Finally, as a consequence of the work of de Rham, we would now say that the
differential forms form a chain complex, that is, a sequence of abelian groups {Ωk | k ∈
N } and homomorphisms dk : Ωk → Ωk+1 (k ∈ N) with dk+1 ◦ dk = 0. In the language
of differential forms used by physicists and engineers: “Every exact form is closed”.
[Recall that the differential k-form ω is called closed if dk ω = 0 and it is called exact if
it can be written as dk−1 σ for some (k − 1)-form σ.]
In the case of R3 , d1 ◦d0 = 0 is the familiar statement from vector calculus “curl-grad
= 0” or ∇ × ∇f = 0, and d2 ◦ d1 = 0 is the statement “div-curl = 0” or ∇ · ∇ × v = 0.
Heuristically, the Fundamental Theorem of Calculus and its generalisations in Equation 6 state that the de Rahm complex is dual to the simplicial complex of suitable
spaces.
The k-th homology group of a chain complex is the kernel of the k-th homomorphism
factored by the image of the preceding one, so that
H k (Rn ) := ker dk /imdk−1 .
(9)
It provides a measure of the extent to which closed k-forms are exact. The Fundamental
Theorem of Calculus yields the statement
H 1 (R) = 0.
(10)
It and the more general versions together yield
H k (Rn ) = 0
(11)
for all n, k ∈ N with k > 0.
This purely algebraic statement lands us where mathematics has progressed since
Euler. The increased abstraction has necessitated and been made possible by algebraisation and the development of new, abstract and often non-quantitative theories such
as the modern theories of abstract algebra, functional analysis, integration theory, harmonic analysis, axiomatic set theory, algebraic topology, differential geometry, algebraic
geometry and category theory.
Rather than making modern mathematics more remote from applications and applicability, this abstraction has had the opposite effect: Mathematics is now applied to
more disciplines than ever before, with previously intractable problems now solved. Fermat’s Last Theorem furnishes the most recent example of a problem elementary enough
for a layman to understand, whose solution has required the application of some of the
most abstract mathematical theories, which, on the face of it, have no bearing on the
problem and in ways which few had envisaged.
Moreover, because of this algebraisation has meant many calculations which formerly required skill and/or ingenuity have been reduced to routine manipulations and
algorithmic procedures, suited to solution by computers.
3
How Does This Affect the Way We Should Teach
Mathematics?
It would be patently absurd to appeal to de Rham cohomology in a first course on
calculus! But it is equally absurd to present such a course as if de Rham cohomology
belonged to the realm of science fiction.
The principal difficulty with many current “elementary” and “introductory” courses
is that by hiding — or, at best, just ignoring — rather than revealing and emphasising
the unity and coherence of modern mathematics, they impede both understanding and
subsequent advancement.
Mathematics courses, especially “low level” ones, are all too frequently taught as
collections of computational tricks and arcane formulæ to be remembered just long
enough to pass the next examination.
Students often complain of needing to forget pictures and “intuitions” acquired in
first calculus courses when learning multivariate calculus and having to start all over
again.
Students typically perceive calculus and algebra as mutually exclusive, as must be
expected from the way they are usually taught.
I believe that calculus is best taught algebraically, given the historical development
outlined very briefly above. That history actually illustrates two different ways in which
the algebraisation of mathematics in general, but especially calculus, has occurred.
Firstly, it has become clear that calculus is the study of F(X), the real algebra of
real valued functions on the subset X of Rn and of certain of its sub-algebras.
Secondly — and this is the more recent development — we now assign to each of the
subsets X of Rn a collection of algebraic invariants which reflect many of the significant
properties of X.
Clearly this second aspect can hardly be introduced into a first course on calculus,
although it might be alluded to in informal discussion of what a student would meet
eventually, when pursuing mathematics far enough.
But I have tried basing the calculus courses on the first aspect, viewing calculus as
the study of the real algebra, F(X), of real valued functions on the subset X of Rn
and of certain of its sub-algebras. A first course in calculus typically considers only the
case n = 1 with X ⊆ R. It is the central notion of a limit which distinguishes calculus
from “pure” algebra and calculus may be regarded fruitfully as the study of how limits
interact with the algebraic structure of F(X).
This has the immediate pedagogical advantage of providing context for the central
theorems, thereby dispelling much of the mysteriousness and lack of motivation students
so often criticise. Moreover it leads naturally to other topics. Differential geometry, for
example, may be fruitfully regarded as the study of the sub-algebra, C ∞ (X), of F(X)
consisting of all smooth real-valued functions defined on the manifold X. While this
seems to lack geometry, it is an amazing fact that under mild conditions, the manifold
can be recovered from the algebraic structure of C ∞ (X)! Of course computation in
differential geometry relies on the algebra.
It also demonstrates that much of the work can be reduced to “mere” symbolic
computation and done by machine, the crucial step being the determination of how
limits behave and how they interact with the algebraic operations on F(X).
Of course, I do not commence teaching calculus by announcing to students that we
shall be spending the course studying the real algebra F(X), just as I would definitely
not say to pupils at the outset of learning elementary arithmetic at primary school that
they will be learning about the ring of integers!
Rather, I introduce the structure piece by piece, establishing the relevant properties.
The course is guided by the algebraic structure, so that after the course, a student
meeting the notion of an algebra and homomorphism of algebras can readily recognise
that the calculus course provided examples of algebras and homomorphisms between
them. After all, upon being told what a ring is, any pupil who has a good mastery of
elementary arithmetic should appreciate that the integers form a ring.
Specifically, we define a sum and product on the elements of F(X) “point-wise”,
that is to say, given functions f, g : X −→ R, we define
f + g : X −→ R,
f.g : X −→ R,
x 7−→ f (x) + g(x)
x 7−→ f (x).g(x)
(12)
(13)
At this stage it can be fruitfully pointed out that in fact this also includes multiplying
a function by a real constant, for we may identify each real number c with the constant
function X −→ R which assigns c to each and every element of X. In this way we
may regard F(X) as an extension of the set of real numbers, very much the way that
the integers form an extension of the counting numbers, the rational numbers of the
integers and the real numbers of the rational ones.
If the range of g is a subset of X, we can also define the composition of g and f ,
f ◦ g : X −→ R,
x 7−→ f (g(x)).
(14)
These operations provide the algebraic structure on F(X). Applying them to the
functions
(i) ec : X −→ R,
x 7−→ c
(ii) p1 : X −→ R,
x 7−→ x
is sufficient to define all polynomial functions on X.
If we add
(iii) s : X −→ R,
(iv) p−1 : X −→ R,
x 7−→ sin x
x 7−→
1
x
as long as 0 ∈
/ X,
we have all the rational functions and trigonometric functions.
Finally if we include
(v) exp : X −→ R,
x 7−→ ex ,
together with the inverse functions to all of the above whenever these are defined, then
each elementary function — and therefore all those studied in calculus courses — arise
by means of recursion. In other words, these few “basic” functions, together with the
three algebraic operations generate all the functions met in a first calculus course. These
functions are frequently referred to as elementary functions and we write E(X) for the
set of all elementary real valued functions defined on X. Clearly, E(X) is a sub-algebra
of F(X).
It is therefore enough to study the behaviour of these five functions as long as we
know how the algebraic operations behave with respect to the other operations studied
in calculus.
The behaviour of our basic functions with respect to taking limits is easily determined directly “from first principles” in the case of the first three and with appeal to
the properties of the functions in the case of the last two.
As to the relationship between limits and our algebraic operations, it is easy to show
that
lim (f + g)(x) =
lim f (x) + lim g(x)
(15)
lim (f.g)(x) = [ lim f (x)].[ lim g(x)],
(16)
x→a
x→a
x→a
x→a
x→a
x→a
as long as both lim f (x) and lim g(x) exist.
x→a
x→a
Moreover, if lim g(x) = k and lim f (y) = `, then
x→a
y→k
lim (f ◦ g)(x) = `.
(17)
x→a
Of course we must specialise to a sub-algebra of F(X) to ensure that the limits
exist and this again illustrates a common procedure in mathematics: When something
does not work in complete generality, restrict attention to the cases where it does and
investigate the minimal restrictions required as well as the reason(s) for the failure in
complete generality.
Furthermore, this leads to a very natural way of distinguishing continuous functions,
for these are precisely the functions for which we can evaluate lim f (x) by “plugging
x→a
in”, that is, by evaluating f at a. Such functions form a sub-algebra, C 0 (X), of F(X).
The corresponding results in the case of differentiation are no more difficult in the
case of our basic functions, and the rules for the behaviour of differentiation with respect
to the algebraic operations are precisely the linearity, Leibniz rule and chain rule every
student meets. Once again we must restrict the set of admissible functions to which we
can apply this operation.
We have the following table for this more restricted set of functions.
Limits
linearity
rule for products
rule for composites
Differentiation
linearity
Leibniz rule
chain rule
We can therefore evaluate limits, differentiate explicitly any function which is generated by our five basic functions using the three algebraic operations. Thus we have,
in effect, an algorithm for evaluating the limits of, or equally for differentiating, the
class of functions generated by our basic functions and the algebraic operations:
1. Express the function in terms of our basic functions using only the three algebraic
operations.
2. Write down the limit/derivative of each of the basic functions appearing.
3. Execute the algebraic operation on the appropriate limit/derivative corresponding
to each algebraic operation on the functions.
It “only” remains to translate this into your favourite programming language.
The case of integration is more interesting. Our basic functions can be integrated
directly with ease and, using the Fundamental Theorem of Calculus, we derive the linearity of the integral, integration by parts and integration by substitution corresponding
to our three algebraic operations.
We can tabulate the relationships as follows.
Limits
linearity
rule for products
rule for composites
Differentiation
linearity
Leibniz rule
chain rule
Integration
linearity
integration by parts
integration by substitution
But there are differences as well between differentiation and integration.
Whereas we specialised as we went from just functions to continuous functions and
again when we went from continuous functions to differentiable ones, we do not continue
this line of specialisation by passing to integrable functions. On the contrary, while
every continuous function (and a fortiori every differentiable one) is integrable, not
every integrable function is continuous.
Moreover difficulties arise if we insist on explicitly expressing integrals purely in
terms of our basic functions and the three algebraic operations. For while differentiation
maps our special subset, E(X), of F(X) to itself, integration does not. The above
algorithm cannot be used to calculate integrals instead of taking limits or calculating
derivatives.
This inconvenience has several didactic advantages.
a. It provides a natural example of a problem which cannot be solved algorithmically.
b. It naturally raises questions leading to further and deeper study of mathematics
and opens the way to more applications. It provides, for example, motivation for
Taylor series and Fourier series as techniques for evaluating otherwise inaccessible
integrals by using readily computable ones and limits, demonstrating once more
the power of the central notion of calculus.
c. It illustrates another situation common in mathematics: We push our available
techniques as far as we can and then seek to find other means when we are confronted with situations beyond the scope of our current techniques and methods.
Moreover, the obstacles and difficulties we meet often prescribe specifications for
the new techniques and methods we need to develop.
What more can we wish for as teachers of mathematics?
References
[1] Bott, R., Tu, L.W., Differential Forms in Algebraic Topology, Springer-Verlag.
Graduate Texts in Mathematics 82, 1982.
[2] Dieudonn´e, J., A Panorama of Pure Mathematics, Academic Press, 1982.
[3] James, I.M. (ed.), History of Topology, North-Holland 1999.